Knabner, Peter
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Knabner, Peter
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Knabner, Peter
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Item-typ:Veröffentlichung, Two-scale models for reactive transport and evolving microstructure(2008-06-26); ; ; Reactive transport in materials with a complex microstructure is a phenomenon that frequently occurs in nature and in technical applications. An example are chemical reactions taking place in the pores of concrete structures. Since these processes are highly influenced by the geometry of the microstructure, it is a difficult task to understand and predict their behaviour on a macroscopic scale. It has been shown by periodic homogenisation that for certain situations two-scale models are appropriate. Periodic homogenisation is an averaging method that is limited to a uniform and constant microstructure. However, in general a porous medium contains multiple phases that evolve in time. For instance, the air-water mixture in an unsaturated porous medium varies due to chemical reactions or non-constant boundary conditions. In certain applications, such effects need to be captured in a mathematical model. In this thesis, two-scale models are formulated for reactive transport in a porous medium with an evolving phase configuration. The reaction-diffusion problem is motivated by a chemical degradation process in concrete structures. The model equations are formally derived from a pore-scale model by homogenisation, but are defined on a time-dependent geometry. Two different settings are considered: In the first one, the evolution of the pore geometry is independent of the reaction-diffusion process and can be considered as a-priori given. This leads to a semilinear two-scale system of parabolic PDEs. In the second setting, the pore water evolves due to the reaction itself, which results in a quasilinear model. Using a transformation to a reference configuration, it is proven that both two-scale models are well-posed. A simple numerical implementation that works with both settings is suggested and validated. The qualitative behaviour of the solutions of the two-scale models are illustrated for an academic example. In summary, the results give a solid theoretical fundament for further important tasks such as the identification of model parameters or the design of efficient numerical algorithms suitable for real-world applications.Dissertation437 539 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an H^{1,p} Setting(2013-05-17); ; ; The processes of chemical transport in porous media are extensively studied in the fields of applied mathematics, material science, chemical engineering etc. A porous medium (e.g. concrete, soil, rocks, reservoir etc.) is a multiscale material/medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed by the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods. The transport process in a porous medium is a complex phenomena. In this thesis, the heterogeneities inside a porous medium are assumed to be periodically distributed and diffusion-reaction of a finite number of chemical species are investigated. Two different models are proposed in this work. In model M1, diffusion-reaction of mobile chemical species are considered. The chemical processes are modeled via mass action kinetics and the modeling leads to a system of multi-species diffusion-reaction equations (nonlinear partial differential equations) at the micro scale. For this system of equations, existence of a unique positive global weak solution is proved by the help of a Lyapunov functional and Schaefer's fixed point theorem. The upscaled model of this system is obtained using periodic homogenization which is an averaging method. In model M2, we consider diffusion-advection-reaction of two different types of mobile species (type I and type II). The type II species are supplied via dissolution process due to the presence of immobile species on the surface of the solid parts. The presence of mobile and the immobile species make the model complex and the modeling yields a coupled system of nonlinear partial differential equations. The existence of a unique positive global weak solution of this complex system is shown. Finally, with the help of periodic homogenization, model M2 is upscaled from the micro scale to the macro scale. Numerical simulations are conducted for both models separately. For the purpose of illustration, we restrict ourselves to relatively simple 2-dimensional situations. For models M1 and M2, simulation results at the micro scale and at the macro scale are compared.Dissertation328 194